Borsuk's Cohomotopy Groups

Abstract

In 1936 Borsuk2 [2] indicated how the homotopy classes of maps of a space X into a space Y form a group under rather general conditions. In particular, if Y is an n-sphere and X is a space with dim X 2. The first step in the classification is the setting up of a map : -*(K > Hn(Kn+l) where rfn(Kfn+l) is the set of homotopy classes in question, while H n(I n+) is the nth integral cohomology group of Kn+1. It is relatively easy to show that f is a mapping onto. The next and much harder step is to study the sets f-`(u) for u E Hn(Kn+l). Here there is a marked difference between the cases n = 2 and n > 2. For n = 2 the structure (and even the cardinal number) of J7'(u) depends on the choice of u. For n > 2 all the sets 7 l(u) are in 1-1 correspondence with a certain factor group of the cohomology group H n+(K n+; 2) with mod 2 coefficients. This suggests that for n > 2 the set 7rf(Kf+l) could be given a group structure in such a way that f is a homomorphism with respect to this group structure. The main part of Steenrod's classification theorem would be the description of the kernel of I. It is shown herein that Steenrod's theorem exhibits the group Xrn(Kn+1) as a group extension of a calculable group3 by another calculable group. Hence, the original question suggested by Steenrod's results is answered. Furthermore,